Simulation of a single-bubble process

The following function generates a time series which switches from a martingale to a mildly explosive process and then back to a martingale.

Usage

sim_psy1(
  n,
  te = 0.4 * n,
  tf = 0.15 * n + te,
  c = 1,
  alpha = 0.6,
  sigma = 6.79,
  seed = NULL,
  e = NULL,
  shifts = NULL,
  coef_noise = NULL,
  coef_a = 1
)

Arguments

n: A positive integer specifying the length of the simulated output series.
te: A scalar in (0, tf) specifying the observation in which the bubble originates.
tf: A scalar in (te, n) specifying the observation in which the bubble collapses.
c: A positive scalar determining the autoregressive coefficient in the explosive regime.
alpha: A positive scalar in (0, 1) determining the value of the expansion rate in the autoregressive coefficient.
sigma: A positive scalar indicating the standard deviation of the innovations.
seed: An object specifying if and how the random number generator (rng) should be initialized. Either NULL or an integer will be used in a call to set.seed before simulation. If set, the value is saved as "seed" attribute of the returned value. The default, NULL, will not change rng state, and return .Random.seed as the "seed" attribute. Results are reproducible across the parallel and non-parallel option when the same seed is used.
e: An optional numeric vector of length n - 1 of innovations to use in place of rnorm(n - 1, sd = sigma). Lets the plain PSY equation above be driven by a non-Gaussian/heteroskedastic/dependent shock sequence instead of i.i.d. Gaussian noise – see sim_innov (heavy-tailed/skewed), sim_vol_garch (GARCH/TGARCH), sim_vol_cir/sim_vol_sv (stochastic volatility) and sim_fi (long-memory) for ready-made generators. Default NULL reproduces the plain i.i.d. Gaussian DGP exactly.
shifts: An optional data frame/list with integer element/column date (in 2:n) and numeric element/column size, adding a one-period deterministic level shift of magnitude size at each date – Harvey, Leybourne, Tatlow & Zu (2025)'s level-shift DGP. Default NULL adds no shifts.
coef_noise: An optional numeric vector of length n - 1, mean-zero/unit-variance, perturbing the explosive-regime coefficient as delta + coef_a * coef_noise[t] / sqrt(n) instead of the fixed delta – Kurozumi & Nishi (2025)'s stochastically varying explosive coefficient. Default NULL keeps delta fixed.
coef_a: A positive scalar scaling coef_noise. Ignored if coef_noise is NULL.

Value

A numeric vector of length n.

Details

The data generating process is described by the following equation: $$X_t = X_{t-1}1\{t < \tau_e\}+ \delta_T X_{t-1}1\{\tau_e \leq t\leq \tau_f\} + \left(\sum_{k=\tau_f+1}^t \epsilon_k + X_{\tau_f}\right) 1\{t > \tau_f\} + \epsilon_t 1\{t \leq \tau_f\} $$

where the autoregressive coefficient $\delta_T$ is given by:

$$\delta_T = 1 + cT^{-a}$$

with $c>0$, $\alpha \in (0,1)$, $\epsilon \sim iid(0, \sigma^2)$ and $X_{\tau_f} = X_{\tau_e} + X'$ with $X' = O_p(1)$, $\tau_e = [T r_e]$ dates the origination of the bubble, and $\tau_f = [T r_f]$ dates the collapse of the bubble. During the pre- and post- bubble periods, $[1, \tau_e)$, $X_t$ is a pure random walk process. During the bubble expansion period $\tau_e, \tau_f]$ becomes a mildly explosive process with expansion rate given by the autoregressive coefficient $\delta_T$; and, finally during the post-bubble period, $(\tau_f, \tau]$ $X_t$ reverts to a martingale.

For further details see Phillips et al. (2015) p. 1054.

References

Phillips, P. C. B., Shi, S., & Yu, J. (2015). Testing for Multiple Bubbles: Historical Episodes of Exuberance and Collapse in the S&P 500. International Economic Review, 5 6(4), 1043-1078.

Examples

# 100 periods with bubble origination date 40 and termination date 55
sim_psy1(n = 100, seed = 123) %>%
  autoplot()


# 200 periods with bubble origination date 80 and termination date 110
sim_psy1(n = 200, seed = 123) %>%
  autoplot()


# 200 periods with bubble origination date 100 and termination date 150
sim_psy1(n = 200, te = 100, tf = 150, seed = 123) %>%
  autoplot()


# Same DGP, driven by GARCH(1,1) innovations instead of i.i.d. Gaussian
sim_psy1(n = 200, seed = 123, e = sim_vol_garch(199, seed = 123)) %>%
  autoplot()


# Same DGP, with two deterministic level shifts
sim_psy1(n = 200, seed = 123, shifts = list(date = c(50, 150), size = c(20, -20))) %>%
  autoplot()